# Questions tagged [modular-curves]

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14
questions

**3**

votes

**1**answer

225 views

### Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)

In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer):
''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...

**5**

votes

**1**answer

168 views

### Geometry of the section $X_0(N) \to X_0(pN)$ given by the canonical subgroup

Goren and Kassaei's paper "The Canonical Subgroup: a Subgroup-Free Approach" takes the position that the canonical subgroup of order $p$ for elliptic curves over $\mathbb{Z}_p$ with $\Gamma$-level ...

**2**

votes

**0**answers

179 views

### Standard application of Oort-Tate classification theorem

$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...

**2**

votes

**0**answers

110 views

### Weighted projective lines and elliptic curves

The modular curves of low level can sometimes be describes as weighted projective lines. For example, over $\mathbb{Z}[1/2]$ the compactified stack of elliptic curves with full level 2 structure is ...

**3**

votes

**1**answer

223 views

### Automorphisms of the modular curve defined over $\mathbb{Q}$

Let $p\geq 3$ be a prime number. The modular curve $X(p)$ can be considered as a connected smooth projective curve over complex numbers. There is a subgroup $\mathrm{PSL}_2(\mathbb{F}_p)$ inside its ...

**1**

vote

**1**answer

179 views

### When is $X_0(N)$ representable?

Fix a base ring $R$. Is the rigidification of the modular "curve" $X_0(N)$ in the sense of Abramovich-Olsson-Vistoli representable iff $N\geq 5$ and $N$ is $0, 2, 3, 6, 8, 11 (\mathrm{mod}\: 12)$? ...

**0**

votes

**0**answers

70 views

### Isomorphism between 2nd symmetric product and Jacobian

Let $X=X_0(N)$ be hyperelliptic with $g(X)\geq 2$ with $\infty$ as a cusp and $\iota$ as the hyperelliptic involution. Then the map
$$X^{(2)} \longrightarrow Jac(X)$$ $$D \longrightarrow [D-\infty -\...

**2**

votes

**0**answers

132 views

### What is the modular curve for level 1, 2?

An elliptic curve over a scheme $S$ is the data of a proper smooth morphism
of schemes $\pi: E\rightarrow S$ whose geometric fibres are connected curves of genus $1$ and a section $e: S \rightarrow E$....

**4**

votes

**1**answer

293 views

### When is the conductor of an elliptic modular curve equal to its level?

Suppose the usual modular curve $E=X_0(N)$ over $\mathbb{Q}$ has genus 1 (e.g. $N=15$). Define the conductor of $E/\mathbb{Q}$ as the ideal/integer:
$$M=\prod_{p}p^{f(E/\mathbb{Q}_p)},$$
where
$$f(...

**2**

votes

**0**answers

126 views

### Local-global compatibility and modular curves

I have been told by some people that local-global Langlands compatibility for $GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture ...

**6**

votes

**0**answers

330 views

### Integral models of perfectoid modular curves

Scholze constructed perfectoid modular curve and its canonical and anticanonical part in his paper On torsion in the cohomology of locally symmetric varieties (Annals of Mathematics 182 (2015) pp 945–...

**2**

votes

**0**answers

152 views

### Igusa curve at infinite level

In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....

**7**

votes

**3**answers

446 views

### Endomorphism ring of $J_0(p)$ and Hecke operators

Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ ...

**10**

votes

**0**answers

246 views

### Generalized Jacobians and modular units

Let $X$ be a proper algebraic smooth curve over a characteristic zero field $k$ and let $J$ be the Jacobian variety of $X$. Let $K$ be the function field of $X$. Assume that we are given $n$ distinct ...